Provided by: liblapack-doc_3.11.0-2_all 
NAME
complex16GEcomputational - complex16
SYNOPSIS
Functions
subroutine zgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
ZGEBAK
subroutine zgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
ZGEBAL
subroutine zgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
subroutine zgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
ZGEBRD
subroutine zgecon (NORM, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
ZGECON
subroutine zgeequ (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
ZGEEQU
subroutine zgeequb (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
ZGEEQUB
subroutine zgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO)
ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked
algorithm.
subroutine zgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZGEHRD
subroutine zgelq2 (M, N, A, LDA, TAU, WORK, INFO)
ZGELQ2 computes the LQ factorization of a general rectangular matrix using an
unblocked algorithm.
subroutine zgelqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGELQF
subroutine zgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO)
ZGEMQRT
subroutine zgeql2 (M, N, A, LDA, TAU, WORK, INFO)
ZGEQL2 computes the QL factorization of a general rectangular matrix using an
unblocked algorithm.
subroutine zgeqlf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQLF
subroutine zgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK, RWORK, INFO)
ZGEQP3
subroutine zgeqr2 (M, N, A, LDA, TAU, WORK, INFO)
ZGEQR2 computes the QR factorization of a general rectangular matrix using an
unblocked algorithm.
subroutine zgeqr2p (M, N, A, LDA, TAU, WORK, INFO)
ZGEQR2P computes the QR factorization of a general rectangular matrix with non-
negative diagonal elements using an unblocked algorithm.
subroutine zgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQRF
subroutine zgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQRFP
subroutine zgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO)
ZGEQRT
subroutine zgeqrt2 (M, N, A, LDA, T, LDT, INFO)
ZGEQRT2 computes a QR factorization of a general real or complex matrix using the
compact WY representation of Q.
recursive subroutine zgeqrt3 (M, N, A, LDA, T, LDT, INFO)
ZGEQRT3 recursively computes a QR factorization of a general real or complex matrix
using the compact WY representation of Q.
subroutine zgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR,
WORK, RWORK, INFO)
ZGERFS
subroutine zgerfsx (TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R, C, B, LDB, X, LDX,
RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
INFO)
ZGERFSX
subroutine zgerq2 (M, N, A, LDA, TAU, WORK, INFO)
ZGERQ2 computes the RQ factorization of a general rectangular matrix using an
unblocked algorithm.
subroutine zgerqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGERQF
subroutine zgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, CWORK, LWORK, RWORK,
LRWORK, INFO)
ZGESVJ
subroutine zgetf2 (M, N, A, LDA, IPIV, INFO)
ZGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting
with row interchanges (unblocked algorithm).
subroutine zgetrf (M, N, A, LDA, IPIV, INFO)
ZGETRF
recursive subroutine zgetrf2 (M, N, A, LDA, IPIV, INFO)
ZGETRF2
subroutine zgetri (N, A, LDA, IPIV, WORK, LWORK, INFO)
ZGETRI
subroutine zgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZGETRS
subroutine zhgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z,
LDZ, WORK, LWORK, RWORK, INFO)
ZHGEQZ
subroutine zla_geamv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZLA_GEAMV computes a matrix-vector product using a general matrix to calculate error
bounds.
double precision function zla_gercond_c (TRANS, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY,
INFO, WORK, RWORK)
ZLA_GERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for
general matrices.
double precision function zla_gercond_x (TRANS, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK,
RWORK)
ZLA_GERCOND_X computes the infinity norm condition number of op(A)*diag(x) for general
matrices.
subroutine zla_gerfsx_extended (PREC_TYPE, TRANS_TYPE, N, NRHS, A, LDA, AF, LDAF, IPIV,
COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N, ERRS_C, RES, AYB, DY, Y_TAIL,
RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
ZLA_GERFSX_EXTENDED
double precision function zla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF)
ZLA_GERPVGRW multiplies a square real matrix by a complex matrix.
recursive subroutine zlaqz0 (WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA,
BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, REC, INFO)
ZLAQZ0
subroutine zlaqz1 (ILQ, ILZ, K, ISTARTM, ISTOPM, IHI, A, LDA, B, LDB, NQ, QSTART, Q, LDQ,
NZ, ZSTART, Z, LDZ)
ZLAQZ1
recursive subroutine zlaqz2 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB, Q, LDQ,
Z, LDZ, NS, ND, ALPHA, BETA, QC, LDQC, ZC, LDZC, WORK, LWORK, RWORK, REC, INFO)
ZLAQZ2
subroutine zlaqz3 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS, NBLOCK_DESIRED, ALPHA, BETA,
A, LDA, B, LDB, Q, LDQ, Z, LDZ, QC, LDQC, ZC, LDZC, WORK, LWORK, INFO)
ZLAQZ3
subroutine zlaunhr_col_getrfnp (M, N, A, LDA, D, INFO)
ZLAUNHR_COL_GETRFNP
recursive subroutine zlaunhr_col_getrfnp2 (M, N, A, LDA, D, INFO)
ZLAUNHR_COL_GETRFNP2
subroutine ztgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M,
WORK, RWORK, INFO)
ZTGEVC
subroutine ztgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO)
ZTGEXC
Detailed Description
This is the group of complex16 computational functions for GE matrices
Function Documentation
subroutine zgebak (character JOB, character SIDE, integer N, integer ILO, integer IHI, double
precision, dimension( * ) SCALE, integer M, complex*16, dimension( ldv, * ) V, integer
LDV, integer INFO)
ZGEBAK
Purpose:
ZGEBAK forms the right or left eigenvectors of a complex general
matrix by backward transformation on the computed eigenvectors of the
balanced matrix output by ZGEBAL.
Parameters
JOB
JOB is CHARACTER*1
Specifies the type of backward transformation required:
= 'N': do nothing, return immediately;
= 'P': do backward transformation for permutation only;
= 'S': do backward transformation for scaling only;
= 'B': do backward transformations for both permutation and
scaling.
JOB must be the same as the argument JOB supplied to ZGEBAL.
SIDE
SIDE is CHARACTER*1
= 'R': V contains right eigenvectors;
= 'L': V contains left eigenvectors.
N
N is INTEGER
The number of rows of the matrix V. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
The integers ILO and IHI determined by ZGEBAL.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
SCALE
SCALE is DOUBLE PRECISION array, dimension (N)
Details of the permutation and scaling factors, as returned
by ZGEBAL.
M
M is INTEGER
The number of columns of the matrix V. M >= 0.
V
V is COMPLEX*16 array, dimension (LDV,M)
On entry, the matrix of right or left eigenvectors to be
transformed, as returned by ZHSEIN or ZTREVC.
On exit, V is overwritten by the transformed eigenvectors.
LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgebal (character JOB, integer N, complex*16, dimension( lda, * ) A, integer LDA,
integer ILO, integer IHI, double precision, dimension( * ) SCALE, integer INFO)
ZGEBAL
Purpose:
ZGEBAL balances a general complex matrix A. This involves, first,
permuting A by a similarity transformation to isolate eigenvalues
in the first 1 to ILO-1 and last IHI+1 to N elements on the
diagonal; and second, applying a diagonal similarity transformation
to rows and columns ILO to IHI to make the rows and columns as
close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrix, and improve the
accuracy of the computed eigenvalues and/or eigenvectors.
Parameters
JOB
JOB is CHARACTER*1
Specifies the operations to be performed on A:
= 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
for i = 1,...,N;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If JOB = 'N', A is not referenced.
See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI are set to INTEGER such that on exit
A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N.
SCALE
SCALE is DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied to
A. If P(j) is the index of the row and column interchanged
with row and column j and D(j) is the scaling factor
applied to row and column j, then
SCALE(j) = P(j) for j = 1,...,ILO-1
= D(j) for j = ILO,...,IHI
= P(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N to IHI+1,
then 1 to ILO-1.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The permutations consist of row and column interchanges which put
the matrix in the form
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigenvalues lie
along the diagonal. The column indices ILO and IHI mark the starting
and ending columns of the submatrix B. Balancing consists of applying
a diagonal similarity transformation inv(D) * B * D to make the
1-norms of each row of B and its corresponding column nearly equal.
The output matrix is
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
Information about the permutations P and the diagonal matrix D is
returned in the vector SCALE.
This subroutine is based on the EISPACK routine CBAL.
Modified by Tzu-Yi Chen, Computer Science Division, University of
California at Berkeley, USA
subroutine zgebd2 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16,
dimension( * ) TAUQ, complex*16, dimension( * ) TAUP, complex*16, dimension( * ) WORK,
integer INFO)
ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
Purpose:
ZGEBD2 reduces a complex general m by n matrix A to upper or lower
real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
Parameters
M
M is INTEGER
The number of rows in the matrix A. M >= 0.
N
N is INTEGER
The number of columns in the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the unitary matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the unitary matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors.
See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D
D is DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
E
E is DOUBLE PRECISION array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ
TAUQ is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.
TAUP
TAUP is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.
WORK
WORK is COMPLEX*16 array, dimension (max(M,N))
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
where tauq and taup are complex scalars, and v and u are complex
vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
where tauq and taup are complex scalars, v and u are complex vectors;
v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).
The contents of A on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).
subroutine zgebrd (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16,
dimension( * ) TAUQ, complex*16, dimension( * ) TAUP, complex*16, dimension( * ) WORK,
integer LWORK, integer INFO)
ZGEBRD
Purpose:
ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
bidiagonal form B by a unitary transformation: Q**H * A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
Parameters
M
M is INTEGER
The number of rows in the matrix A. M >= 0.
N
N is INTEGER
The number of columns in the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the unitary matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the unitary matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors.
See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D
D is DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
E
E is DOUBLE PRECISION array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ
TAUQ is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.
TAUP
TAUP is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,M,N).
For optimum performance LWORK >= (M+N)*NB, where NB
is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
where tauq and taup are complex scalars, and v and u are complex
vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
where tauq and taup are complex scalars, and v and u are complex
vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
The contents of A on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).
subroutine zgecon (character NORM, integer N, complex*16, dimension( lda, * ) A, integer LDA,
double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, double
precision, dimension( * ) RWORK, integer INFO)
ZGECON
Purpose:
ZGECON estimates the reciprocal of the condition number of a general
complex matrix A, in either the 1-norm or the infinity-norm, using
the LU factorization computed by ZGETRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).
Parameters
NORM
NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= '1' or 'O': 1-norm;
= 'I': Infinity-norm.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U
as computed by ZGETRF.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
ANORM
ANORM is DOUBLE PRECISION
If NORM = '1' or 'O', the 1-norm of the original matrix A.
If NORM = 'I', the infinity-norm of the original matrix A.
RCOND
RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK
WORK is COMPLEX*16 array, dimension (2*N)
RWORK
RWORK is DOUBLE PRECISION array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgeequ (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
double precision, dimension( * ) R, double precision, dimension( * ) C, double precision
ROWCND, double precision COLCND, double precision AMAX, integer INFO)
ZGEEQU
Purpose:
ZGEEQU computes row and column scalings intended to equilibrate an
M-by-N matrix A and reduce its condition number. R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
R(i) and C(j) are restricted to be between SMLNUM = smallest safe
number and BIGNUM = largest safe number. Use of these scaling
factors is not guaranteed to reduce the condition number of A but
works well in practice.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
The M-by-N matrix whose equilibration factors are
to be computed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
R
R is DOUBLE PRECISION array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A.
C
C is DOUBLE PRECISION array, dimension (N)
If INFO = 0, C contains the column scale factors for A.
ROWCND
ROWCND is DOUBLE PRECISION
If INFO = 0 or INFO > M, ROWCND contains the ratio of the
smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
AMAX is neither too large nor too small, it is not worth
scaling by R.
COLCND
COLCND is DOUBLE PRECISION
If INFO = 0, COLCND contains the ratio of the smallest
C(i) to the largest C(i). If COLCND >= 0.1, it is not
worth scaling by C.
AMAX
AMAX is DOUBLE PRECISION
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= M: the i-th row of A is exactly zero
> M: the (i-M)-th column of A is exactly zero
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgeequb (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
double precision, dimension( * ) R, double precision, dimension( * ) C, double precision
ROWCND, double precision COLCND, double precision AMAX, integer INFO)
ZGEEQUB
Purpose:
ZGEEQUB computes row and column scalings intended to equilibrate an
M-by-N matrix A and reduce its condition number. R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
the radix.
R(i) and C(j) are restricted to be a power of the radix between
SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
of these scaling factors is not guaranteed to reduce the condition
number of A but works well in practice.
This routine differs from ZGEEQU by restricting the scaling factors
to a power of the radix. Barring over- and underflow, scaling by
these factors introduces no additional rounding errors. However, the
scaled entries' magnitudes are no longer approximately 1 but lie
between sqrt(radix) and 1/sqrt(radix).
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
The M-by-N matrix whose equilibration factors are
to be computed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
R
R is DOUBLE PRECISION array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A.
C
C is DOUBLE PRECISION array, dimension (N)
If INFO = 0, C contains the column scale factors for A.
ROWCND
ROWCND is DOUBLE PRECISION
If INFO = 0 or INFO > M, ROWCND contains the ratio of the
smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
AMAX is neither too large nor too small, it is not worth
scaling by R.
COLCND
COLCND is DOUBLE PRECISION
If INFO = 0, COLCND contains the ratio of the smallest
C(i) to the largest C(i). If COLCND >= 0.1, it is not
worth scaling by C.
AMAX
AMAX is DOUBLE PRECISION
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= M: the i-th row of A is exactly zero
> M: the (i-M)-th column of A is exactly zero
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgehd2 (integer N, integer ILO, integer IHI, complex*16, dimension( lda, * ) A,
integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer
INFO)
ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked
algorithm.
Purpose:
ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
by a unitary similarity transformation: Q**H * A * Q = H .
Parameters
N
N is INTEGER
The order of the matrix A. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to ZGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= max(1,N).
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the unitary matrix Q as a product of elementary
reflectors. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU
TAU is COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).
subroutine zgehrd (integer N, integer ILO, integer IHI, complex*16, dimension( lda, * ) A,
integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer
LWORK, integer INFO)
ZGEHRD
Purpose:
ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
an unitary similarity transformation: Q**H * A * Q = H .
Parameters
N
N is INTEGER
The order of the matrix A. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to ZGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the unitary matrix Q as a product of elementary
reflectors. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU
TAU is COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero.
WORK
WORK is COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,N).
For good performance, LWORK should generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).
This file is a slight modification of LAPACK-3.0's ZGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See ZLAHR2.)
subroutine zgelq2 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)
ZGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked
algorithm.
Purpose:
ZGELQ2 computes an LQ factorization of a complex m-by-n matrix A:
A = ( L 0 ) * Q
where:
Q is a n-by-n orthogonal matrix;
L is a lower-triangular m-by-m matrix;
0 is a m-by-(n-m) zero matrix, if m < n.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and below the diagonal of the array
contain the m by min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors (see Further Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(k)**H . . . H(2)**H H(1)**H, where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in
A(i,i+1:n), and tau in TAU(i).
subroutine zgelqf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer
INFO)
ZGELQF
Purpose:
ZGELQF computes an LQ factorization of a complex M-by-N matrix A:
A = ( L 0 ) * Q
where:
Q is a N-by-N orthogonal matrix;
L is a lower-triangular M-by-M matrix;
0 is a M-by-(N-M) zero matrix, if M < N.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the m-by-min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors (see Further Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(k)**H . . . H(2)**H H(1)**H, where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in
A(i,i+1:n), and tau in TAU(i).
subroutine zgemqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer
NB, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldt, * ) T,
integer LDT, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * )
WORK, integer INFO)
ZGEMQRT
Purpose:
ZGEMQRT overwrites the general complex M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q C C Q
TRANS = 'C': Q**H C C Q**H
where Q is a complex orthogonal matrix defined as the product of K
elementary reflectors:
Q = H(1) H(2) . . . H(K) = I - V T V**H
generated using the compact WY representation as returned by ZGEQRT.
Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**H from the Left;
= 'R': apply Q or Q**H from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'C': Conjugate transpose, apply Q**H.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
NB
NB is INTEGER
The block size used for the storage of T. K >= NB >= 1.
This must be the same value of NB used to generate T
in ZGEQRT.
V
V is COMPLEX*16 array, dimension (LDV,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
ZGEQRT in the first K columns of its array argument A.
LDV
LDV is INTEGER
The leading dimension of the array V.
If SIDE = 'L', LDA >= max(1,M);
if SIDE = 'R', LDA >= max(1,N).
T
T is COMPLEX*16 array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by ZGEQRT, stored as a NB-by-N matrix.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= NB.
C
C is COMPLEX*16 array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q C, Q**H C, C Q**H or C Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is COMPLEX*16 array. The dimension of WORK is
N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgeql2 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)
ZGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked
algorithm.
Purpose:
ZGEQL2 computes a QL factorization of a complex m by n matrix A:
A = Q * L.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the m by n lower trapezoidal matrix L;
the remaining elements, with the array TAU, represent the
unitary matrix Q as a product of elementary reflectors
(see Further Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i).
subroutine zgeqlf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer
INFO)
ZGEQLF
Purpose:
ZGEQLF computes a QL factorization of a complex M-by-N matrix A:
A = Q * L.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the M-by-N lower trapezoidal matrix L;
the remaining elements, with the array TAU, represent the
unitary matrix Q as a product of elementary reflectors
(see Further Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i).
subroutine zgeqp3 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
integer, dimension( * ) JPVT, complex*16, dimension( * ) TAU, complex*16, dimension( * )
WORK, integer LWORK, double precision, dimension( * ) RWORK, integer INFO)
ZGEQP3
Purpose:
ZGEQP3 computes a QR factorization with column pivoting of a
matrix A: A*P = Q*R using Level 3 BLAS.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the upper triangle of the array contains the
min(M,N)-by-N upper trapezoidal matrix R; the elements below
the diagonal, together with the array TAU, represent the
unitary matrix Q as a product of min(M,N) elementary
reflectors.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT
JPVT is INTEGER array, dimension (N)
On entry, if JPVT(J).ne.0, the J-th column of A is permuted
to the front of A*P (a leading column); if JPVT(J)=0,
the J-th column of A is a free column.
On exit, if JPVT(J)=K, then the J-th column of A*P was the
the K-th column of A.
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors.
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO=0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= N+1.
For optimal performance LWORK >= ( N+1 )*NB, where NB
is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
RWORK
RWORK is DOUBLE PRECISION array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a real/complex vector
with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in TAU(i).
Contributors:
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer
Science Dept., Duke University, USA
subroutine zgeqr2 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)
ZGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked
algorithm.
Purpose:
ZGEQR2 computes a QR factorization of a complex m-by-n matrix A:
A = Q * ( R ),
( 0 )
where:
Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix;
0 is a (m-n)-by-n zero matrix, if m > n.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors (see Further Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
subroutine zgeqr2p (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)
ZGEQR2P computes the QR factorization of a general rectangular matrix with non-negative
diagonal elements using an unblocked algorithm.
Purpose:
ZGEQR2P computes a QR factorization of a complex m-by-n matrix A:
A = Q * ( R ),
( 0 )
where:
Q is a m-by-m orthogonal matrix;
R is an upper-triangular n-by-n matrix with nonnegative diagonal
entries;
0 is a (m-n)-by-n zero matrix, if m > n.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n). The diagonal entries of R
are real and nonnegative; the elements below the diagonal,
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors (see Further Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
See Lapack Working Note 203 for details
subroutine zgeqrf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer
INFO)
ZGEQRF
Purpose:
ZGEQRF computes a QR factorization of a complex M-by-N matrix A:
A = Q * ( R ),
( 0 )
where:
Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the unitary matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
subroutine zgeqrfp (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer
INFO)
ZGEQRFP
Purpose:
ZGEQR2P computes a QR factorization of a complex M-by-N matrix A:
A = Q * ( R ),
( 0 )
where:
Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix with nonnegative diagonal
entries;
0 is a (M-N)-by-N zero matrix, if M > N.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n). The diagonal entries of R
are real and nonnegative; The elements below the diagonal,
with the array TAU, represent the unitary matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
See Lapack Working Note 203 for details
subroutine zgeqrt (integer M, integer N, integer NB, complex*16, dimension( lda, * ) A,
integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * )
WORK, integer INFO)
ZGEQRT
Purpose:
ZGEQRT computes a blocked QR factorization of a complex M-by-N matrix A
using the compact WY representation of Q.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
NB
NB is INTEGER
The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if M >= N); the elements below the diagonal
are the columns of V.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is COMPLEX*16 array, dimension (LDT,MIN(M,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See below
for further details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= NB.
WORK
WORK is COMPLEX*16 array, dimension (NB*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A.
Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each
block is of order NB except for the last block, which is of order
IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
for the last block) T's are stored in the NB-by-K matrix T as
T = (T1 T2 ... TB).
subroutine zgeqrt2 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( ldt, * ) T, integer LDT, integer INFO)
ZGEQRT2 computes a QR factorization of a general real or complex matrix using the compact
WY representation of Q.
Purpose:
ZGEQRT2 computes a QR factorization of a complex M-by-N matrix A,
using the compact WY representation of Q.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= N.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the complex M-by-N matrix A. On exit, the elements on and
above the diagonal contain the N-by-N upper triangular matrix R; the
elements below the diagonal are the columns of V. See below for
further details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is COMPLEX*16 array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A. The
block reflector H is then given by
H = I - V * T * V**H
where V**H is the conjugate transpose of V.
recursive subroutine zgeqrt3 (integer M, integer N, complex*16, dimension( lda, * ) A, integer
LDA, complex*16, dimension( ldt, * ) T, integer LDT, integer INFO)
ZGEQRT3 recursively computes a QR factorization of a general real or complex matrix using
the compact WY representation of Q.
Purpose:
ZGEQRT3 recursively computes a QR factorization of a complex M-by-N
matrix A, using the compact WY representation of Q.
Based on the algorithm of Elmroth and Gustavson,
IBM J. Res. Develop. Vol 44 No. 4 July 2000.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= N.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the complex M-by-N matrix A. On exit, the elements on
and above the diagonal contain the N-by-N upper triangular matrix R;
the elements below the diagonal are the columns of V. See below for
further details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is COMPLEX*16 array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A. The
block reflector H is then given by
H = I - V * T * V**H
where V**H is the conjugate transpose of V.
For details of the algorithm, see Elmroth and Gustavson (cited above).
subroutine zgerfs (character TRANS, integer N, integer NRHS, complex*16, dimension( lda, * )
A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * )
IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X,
integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR,
complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)
ZGERFS
Purpose:
ZGERFS improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates for
the solution.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
The original N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by ZGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from ZGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
B
B is COMPLEX*16 array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is COMPLEX*16 array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by ZGETRS.
On exit, the improved solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR
FERR is DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR
BERR is DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK
WORK is COMPLEX*16 array, dimension (2*N)
RWORK
RWORK is DOUBLE PRECISION array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgerfsx (character TRANS, character EQUED, integer N, integer NRHS, complex*16,
dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF,
integer, dimension( * ) IPIV, double precision, dimension( * ) R, double precision,
dimension( * ) C, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension(
ldx , * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) BERR,
integer N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double
precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double precision,
dimension( * ) PARAMS, complex*16, dimension( * ) WORK, double precision, dimension( * )
RWORK, integer INFO)
ZGERFSX
Purpose:
ZGERFSX improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates
for the solution. In addition to normwise error bound, the code
provides maximum componentwise error bound if possible. See
comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
error bounds.
The original system of linear equations may have been equilibrated
before calling this routine, as described by arguments EQUED, R
and C below. In this case, the solution and error bounds returned
are for the original unequilibrated system.
Some optional parameters are bundled in the PARAMS array. These
settings determine how refinement is performed, but often the
defaults are acceptable. If the defaults are acceptable, users
can pass NPARAMS = 0 which prevents the source code from accessing
the PARAMS argument.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)
EQUED
EQUED is CHARACTER*1
Specifies the form of equilibration that was done to A
before calling this routine. This is needed to compute
the solution and error bounds correctly.
= 'N': No equilibration
= 'R': Row equilibration, i.e., A has been premultiplied by
diag(R).
= 'C': Column equilibration, i.e., A has been postmultiplied
by diag(C).
= 'B': Both row and column equilibration, i.e., A has been
replaced by diag(R) * A * diag(C).
The right hand side B has been changed accordingly.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
The original N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by ZGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from ZGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
R
R is DOUBLE PRECISION array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is
multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
is not accessed.
If R is accessed, each element of R should be a power of the radix
to ensure a reliable solution and error estimates. Scaling by
powers of the radix does not cause rounding errors unless the
result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.
C
C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If EQUED = 'C' or 'B', A is
multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
is not accessed.
If C is accessed, each element of C should be a power of the radix
to ensure a reliable solution and error estimates. Scaling by
powers of the radix does not cause rounding errors unless the
result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.
B
B is COMPLEX*16 array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X
X is COMPLEX*16 array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by ZGETRS.
On exit, the improved solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND
RCOND is DOUBLE PRECISION
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.
BERR
BERR is DOUBLE PRECISION array, dimension (NRHS)
Componentwise relative backward error. This is the
componentwise relative backward error of each solution vector X(j)
(i.e., the smallest relative change in any element of A or B that
makes X(j) an exact solution).
N_ERR_BNDS
N_ERR_BNDS is INTEGER
Number of error bounds to return for each right hand side
and each type (normwise or componentwise). See ERR_BNDS_NORM and
ERR_BNDS_COMP below.
ERR_BNDS_NORM
ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
ERR_BNDS_COMP
ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.
NPARAMS
NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS. If <= 0, the
PARAMS array is never referenced and default values are used.
PARAMS
PARAMS is DOUBLE PRECISION array, dimension NPARAMS
Specifies algorithm parameters. If an entry is < 0.0, then
that entry will be filled with default value used for that
parameter. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not.
Default: 1.0D+0
= 0.0: No refinement is performed, and no error bounds are
computed.
= 1.0: Use the double-precision refinement algorithm,
possibly with doubled-single computations if the
compilation environment does not support DOUBLE
PRECISION.
(other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement.
Default: 10
Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm. Positive
is true, 0.0 is false.
Default: 1.0 (attempt componentwise convergence)
WORK
WORK is COMPLEX*16 array, dimension (2*N)
RWORK
RWORK is DOUBLE PRECISION array, dimension (2*N)
INFO
INFO is INTEGER
= 0: Successful exit. The solution to every right-hand side is
guaranteed.
< 0: If INFO = -i, the i-th argument had an illegal value
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
has been completed, but the factor U is exactly singular, so
the solution and error bounds could not be computed. RCOND = 0
is returned.
= N+J: The solution corresponding to the Jth right-hand side is
not guaranteed. The solutions corresponding to other right-
hand sides K with K > J may not be guaranteed as well, but
only the first such right-hand side is reported. If a small
componentwise error is not requested (PARAMS(3) = 0.0) then
the Jth right-hand side is the first with a normwise error
bound that is not guaranteed (the smallest J such
that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
the Jth right-hand side is the first with either a normwise or
componentwise error bound that is not guaranteed (the smallest
J such that either ERR_BNDS_NORM(J,1) = 0.0 or
ERR_BNDS_COMP(J,1) = 0.0). See the definition of
ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
about all of the right-hand sides check ERR_BNDS_NORM or
ERR_BNDS_COMP.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgerq2 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)
ZGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked
algorithm.
Purpose:
ZGERQ2 computes an RQ factorization of a complex m by n matrix A:
A = R * Q.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the m by n upper trapezoidal matrix R; the remaining
elements, with the array TAU, represent the unitary matrix
Q as a product of elementary reflectors (see Further
Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1)**H H(2)**H . . . H(k)**H, where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on
exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
subroutine zgerqf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer
INFO)
ZGERQF
Purpose:
ZGERQF computes an RQ factorization of a complex M-by-N matrix A:
A = R * Q.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
unitary matrix Q as a product of min(m,n) elementary
reflectors (see Further Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1)**H H(2)**H . . . H(k)**H, where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on
exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
subroutine zgesvj (character*1 JOBA, character*1 JOBU, character*1 JOBV, integer M, integer N,
complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( n ) SVA,
integer MV, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( lwork )
CWORK, integer LWORK, double precision, dimension( lrwork ) RWORK, integer LRWORK, integer
INFO)
ZGESVJ
Purpose:
ZGESVJ computes the singular value decomposition (SVD) of a complex
M-by-N matrix A, where M >= N. The SVD of A is written as
[++] [xx] [x0] [xx]
A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
[++] [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N unitary matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.
Parameters
JOBA
JOBA is CHARACTER*1
Specifies the structure of A.
= 'L': The input matrix A is lower triangular;
= 'U': The input matrix A is upper triangular;
= 'G': The input matrix A is general M-by-N matrix, M >= N.
JOBU
JOBU is CHARACTER*1
Specifies whether to compute the left singular vectors
(columns of U):
= 'U' or 'F': The left singular vectors corresponding to the nonzero
singular values are computed and returned in the leading
columns of A. See more details in the description of A.
The default numerical orthogonality threshold is set to
approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=DLAMCH('E').
= 'C': Analogous to JOBU='U', except that user can control the
level of numerical orthogonality of the computed left
singular vectors. TOL can be set to TOL = CTOL*EPS, where
CTOL is given on input in the array WORK.
No CTOL smaller than ONE is allowed. CTOL greater
than 1 / EPS is meaningless. The option 'C'
can be used if M*EPS is satisfactory orthogonality
of the computed left singular vectors, so CTOL=M could
save few sweeps of Jacobi rotations.
See the descriptions of A and WORK(1).
= 'N': The matrix U is not computed. However, see the
description of A.
JOBV
JOBV is CHARACTER*1
Specifies whether to compute the right singular vectors, that
is, the matrix V:
= 'V' or 'J': the matrix V is computed and returned in the array V
= 'A': the Jacobi rotations are applied to the MV-by-N
array V. In other words, the right singular vector
matrix V is not computed explicitly; instead it is
applied to an MV-by-N matrix initially stored in the
first MV rows of V.
= 'N': the matrix V is not computed and the array V is not
referenced
M
M is INTEGER
The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
N
N is INTEGER
The number of columns of the input matrix A.
M >= N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
If JOBU = 'U' .OR. JOBU = 'C':
If INFO = 0 :
RANKA orthonormal columns of U are returned in the
leading RANKA columns of the array A. Here RANKA <= N
is the number of computed singular values of A that are
above the underflow threshold DLAMCH('S'). The singular
vectors corresponding to underflowed or zero singular
values are not computed. The value of RANKA is returned
in the array RWORK as RANKA=NINT(RWORK(2)). Also see the
descriptions of SVA and RWORK. The computed columns of U
are mutually numerically orthogonal up to approximately
TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
see the description of JOBU.
If INFO > 0,
the procedure ZGESVJ did not converge in the given number
of iterations (sweeps). In that case, the computed
columns of U may not be orthogonal up to TOL. The output
U (stored in A), SIGMA (given by the computed singular
values in SVA(1:N)) and V is still a decomposition of the
input matrix A in the sense that the residual
|| A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small.
If JOBU = 'N':
If INFO = 0 :
Note that the left singular vectors are 'for free' in the
one-sided Jacobi SVD algorithm. However, if only the
singular values are needed, the level of numerical
orthogonality of U is not an issue and iterations are
stopped when the columns of the iterated matrix are
numerically orthogonal up to approximately M*EPS. Thus,
on exit, A contains the columns of U scaled with the
corresponding singular values.
If INFO > 0:
the procedure ZGESVJ did not converge in the given number
of iterations (sweeps).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
SVA
SVA is DOUBLE PRECISION array, dimension (N)
On exit,
If INFO = 0 :
depending on the value SCALE = RWORK(1), we have:
If SCALE = ONE:
SVA(1:N) contains the computed singular values of A.
During the computation SVA contains the Euclidean column
norms of the iterated matrices in the array A.
If SCALE .NE. ONE:
The singular values of A are SCALE*SVA(1:N), and this
factored representation is due to the fact that some of the
singular values of A might underflow or overflow.
If INFO > 0:
the procedure ZGESVJ did not converge in the given number of
iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
MV
MV is INTEGER
If JOBV = 'A', then the product of Jacobi rotations in ZGESVJ
is applied to the first MV rows of V. See the description of JOBV.
V
V is COMPLEX*16 array, dimension (LDV,N)
If JOBV = 'V', then V contains on exit the N-by-N matrix of
the right singular vectors;
If JOBV = 'A', then V contains the product of the computed right
singular vector matrix and the initial matrix in
the array V.
If JOBV = 'N', then V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = 'V', then LDV >= max(1,N).
If JOBV = 'A', then LDV >= max(1,MV) .
CWORK
CWORK is COMPLEX*16 array, dimension (max(1,LWORK))
Used as workspace.
If on entry LWORK = -1, then a workspace query is assumed and
no computation is done; CWORK(1) is set to the minial (and optimal)
length of CWORK.
LWORK
LWORK is INTEGER.
Length of CWORK, LWORK >= M+N.
RWORK
RWORK is DOUBLE PRECISION array, dimension (max(6,LRWORK))
On entry,
If JOBU = 'C' :
RWORK(1) = CTOL, where CTOL defines the threshold for convergence.
The process stops if all columns of A are mutually
orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
It is required that CTOL >= ONE, i.e. it is not
allowed to force the routine to obtain orthogonality
below EPSILON.
On exit,
RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
are the computed singular values of A.
(See description of SVA().)
RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero
singular values.
RWORK(3) = NINT(RWORK(3)) is the number of the computed singular
values that are larger than the underflow threshold.
RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi
rotations needed for numerical convergence.
RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
This is useful information in cases when ZGESVJ did
not converge, as it can be used to estimate whether
the output is still useful and for post festum analysis.
RWORK(6) = the largest absolute value over all sines of the
Jacobi rotation angles in the last sweep. It can be
useful for a post festum analysis.
If on entry LRWORK = -1, then a workspace query is assumed and
no computation is done; RWORK(1) is set to the minial (and optimal)
length of RWORK.
LRWORK
LRWORK is INTEGER
Length of RWORK, LRWORK >= MAX(6,N).
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, then the i-th argument had an illegal value
> 0: ZGESVJ did not converge in the maximal allowed number
(NSWEEP=30) of sweeps. The output may still be useful.
See the description of RWORK.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
rotations. In the case of underflow of the tangent of the Jacobi angle, a
modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses
column interchanges of de Rijk [1]. The relative accuracy of the computed
singular values and the accuracy of the computed singular vectors (in
angle metric) is as guaranteed by the theory of Demmel and Veselic [2].
The condition number that determines the accuracy in the full rank case
is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
spectral condition number. The best performance of this Jacobi SVD
procedure is achieved if used in an accelerated version of Drmac and
Veselic [4,5], and it is the kernel routine in the SIGMA library [6].
Some tuning parameters (marked with [TP]) are available for the
implementer.
The computational range for the nonzero singular values is the machine
number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
denormalized singular values can be computed with the corresponding
gradual loss of accurate digits.
Contributor:
============
Zlatko Drmac (Zagreb, Croatia)
References:
[1] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
singular value decomposition on a vector computer.
SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
[2] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
[3] Z. Drmac: Implementation of Jacobi rotations for accurate singular
value computation in floating point arithmetic.
SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
[4] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
LAPACK Working note 169.
[5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
LAPACK Working note 170.
[6] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008, 2015.
Bugs, examples and comments:
===========================
Please report all bugs and send interesting test examples and comments to
drmac@math.hr. Thank you.
subroutine zgetf2 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
integer, dimension( * ) IPIV, integer INFO)
ZGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting
with row interchanges (unblocked algorithm).
Purpose:
ZGETF2 computes an LU factorization of a general m-by-n matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the right-looking Level 2 BLAS version of the algorithm.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, U(k,k) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgetrf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
integer, dimension( * ) IPIV, integer INFO)
ZGETRF
Purpose:
ZGETRF computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the right-looking Level 3 BLAS version of the algorithm.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
recursive subroutine zgetrf2 (integer M, integer N, complex*16, dimension( lda, * ) A, integer
LDA, integer, dimension( * ) IPIV, integer INFO)
ZGETRF2
Purpose:
ZGETRF2 computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the recursive version of the algorithm. It divides
the matrix into four submatrices:
[ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2
A = [ -----|----- ] with n1 = min(m,n)/2
[ A21 | A22 ] n2 = n-n1
[ A11 ]
The subroutine calls itself to factor [ --- ],
[ A12 ]
[ A12 ]
do the swaps on [ --- ], solve A12, update A22,
[ A22 ]
then calls itself to factor A22 and do the swaps on A21.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
IPIV
IPIV is INTEGER array, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row IPIV(i).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgetri (integer N, complex*16, dimension( lda, * ) A, integer LDA, integer,
dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)
ZGETRI
Purpose:
ZGETRI computes the inverse of a matrix using the LU factorization
computed by ZGETRF.
This method inverts U and then computes inv(A) by solving the system
inv(A)*L = inv(U) for inv(A).
Parameters
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the factors L and U from the factorization
A = P*L*U as computed by ZGETRF.
On exit, if INFO = 0, the inverse of the original matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from ZGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO=0, then WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimal performance LWORK >= N*NB, where NB is
the optimal blocksize returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, U(i,i) is exactly zero; the matrix is
singular and its inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgetrs (character TRANS, integer N, integer NRHS, complex*16, dimension( lda, * )
A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer
LDB, integer INFO)
ZGETRS
Purpose:
ZGETRS solves a system of linear equations
A * X = B, A**T * X = B, or A**H * X = B
with a general N-by-N matrix A using the LU factorization computed
by ZGETRF.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U
as computed by ZGETRF.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from ZGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
B
B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zhgeqz (character JOB, character COMPQ, character COMPZ, integer N, integer ILO,
integer IHI, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( ldt, *
) T, integer LDT, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA,
complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldz, * ) Z, integer
LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * )
RWORK, integer INFO)
ZHGEQZ
Purpose:
ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the single-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a complex matrix pair (A,B):
A = Q1*H*Z1**H, B = Q1*T*Z1**H,
as computed by ZGGHRD.
If JOB='S', then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,
H = Q*S*Z**H, T = Q*P*Z**H,
where Q and Z are unitary matrices and S and P are upper triangular.
Optionally, the unitary matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
unitary matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
the matrix pair (A,B) to generalized Hessenberg form, then the output
matrices Q1*Q and Z1*Z are the unitary factors from the generalized
Schur factorization of (A,B):
A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
To avoid overflow, eigenvalues of the matrix pair (H,T)
(equivalently, of (A,B)) are computed as a pair of complex values
(alpha,beta). If beta is nonzero, lambda = alpha / beta is an
eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
A*x = lambda*B*x
and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
alternate form of the GNEP
mu*A*y = B*y.
The values of alpha and beta for the i-th eigenvalue can be read
directly from the generalized Schur form: alpha = S(i,i),
beta = P(i,i).
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
Parameters
JOB
JOB is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Computer eigenvalues and the Schur form.
COMPQ
COMPQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (H,T) is returned;
= 'V': Q must contain a unitary matrix Q1 on entry and
the product Q1*Q is returned.
COMPZ
COMPZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= 'V': Z must contain a unitary matrix Z1 on entry and
the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices H, T, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of H which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H
H is COMPLEX*16 array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = 'S', H contains the upper triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal of H matches that of S, but
the rest of H is unspecified.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max( 1, N ).
T
T is COMPLEX*16 array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T.
On exit, if JOB = 'S', T contains the upper triangular
matrix P from the generalized Schur factorization.
If JOB = 'E', the diagonal of T matches that of P, but
the rest of T is unspecified.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).
ALPHA
ALPHA is COMPLEX*16 array, dimension (N)
The complex scalars alpha that define the eigenvalues of
GNEP. ALPHA(i) = S(i,i) in the generalized Schur
factorization.
BETA
BETA is COMPLEX*16 array, dimension (N)
The real non-negative scalars beta that define the
eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized
Schur factorization.
Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
represent the j-th eigenvalue of the matrix pair (A,B), in
one of the forms lambda = alpha/beta or mu = beta/alpha.
Since either lambda or mu may overflow, they should not,
in general, be computed.
Q
Q is COMPLEX*16 array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the unitary matrix of left Schur
vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
left Schur vectors of (A,B).
Not referenced if COMPQ = 'N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z
Z is COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the unitary matrix of right Schur
vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
RWORK
RWORK is DOUBLE PRECISION array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (H,T) is not
in Schur form, but ALPHA(i) and BETA(i),
i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (H,T) is not
in Schur form, but ALPHA(i) and BETA(i),
i=INFO-N+1,...,N should be correct.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We assume that complex ABS works as long as its value is less than
overflow.
subroutine zla_geamv (integer TRANS, integer M, integer N, double precision ALPHA, complex*16,
dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, integer INCX, double
precision BETA, double precision, dimension( * ) Y, integer INCY)
ZLA_GEAMV computes a matrix-vector product using a general matrix to calculate error
bounds.
Purpose:
ZLA_GEAMV performs one of the matrix-vector operations
y := alpha*abs(A)*abs(x) + beta*abs(y),
or y := alpha*abs(A)**T*abs(x) + beta*abs(y),
where alpha and beta are scalars, x and y are vectors and A is an
m by n matrix.
This function is primarily used in calculating error bounds.
To protect against underflow during evaluation, components in
the resulting vector are perturbed away from zero by (N+1)
times the underflow threshold. To prevent unnecessarily large
errors for block-structure embedded in general matrices,
"symbolically" zero components are not perturbed. A zero
entry is considered "symbolic" if all multiplications involved
in computing that entry have at least one zero multiplicand.
Parameters
TRANS
TRANS is INTEGER
On entry, TRANS specifies the operation to be performed as
follows:
BLAS_NO_TRANS y := alpha*abs(A)*abs(x) + beta*abs(y)
BLAS_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)
BLAS_CONJ_TRANS y := alpha*abs(A**T)*abs(x) + beta*abs(y)
Unchanged on exit.
M
M is INTEGER
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.
Unchanged on exit.
N
N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.
Unchanged on exit.
ALPHA
ALPHA is DOUBLE PRECISION
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
A
A is COMPLEX*16 array, dimension ( LDA, n )
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients.
Unchanged on exit.
LDA
LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, m ).
Unchanged on exit.
X
X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
Before entry, the incremented array X must contain the
vector x.
Unchanged on exit.
INCX
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
BETA
BETA is DOUBLE PRECISION
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.
Unchanged on exit.
Y
Y is DOUBLE PRECISION array, dimension
( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
Before entry with BETA non-zero, the incremented array Y
must contain the vector y. On exit, Y is overwritten by the
updated vector y.
INCY
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit.
Level 2 Blas routine.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zla_gercond_c (character TRANS, integer N, complex*16, dimension(
lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer,
dimension( * ) IPIV, double precision, dimension( * ) C, logical CAPPLY, integer INFO,
complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK)
ZLA_GERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for
general matrices.
Purpose:
ZLA_GERCOND_C computes the infinity norm condition number of
op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate Transpose = Transpose)
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by ZGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by ZGETRF; row i of the matrix was interchanged
with row IPIV(i).
C
C is DOUBLE PRECISION array, dimension (N)
The vector C in the formula op(A) * inv(diag(C)).
CAPPLY
CAPPLY is LOGICAL
If .TRUE. then access the vector C in the formula above.
INFO
INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.
WORK
WORK is COMPLEX*16 array, dimension (2*N).
Workspace.
RWORK
RWORK is DOUBLE PRECISION array, dimension (N).
Workspace.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zla_gercond_x (character TRANS, integer N, complex*16, dimension(
lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer,
dimension( * ) IPIV, complex*16, dimension( * ) X, integer INFO, complex*16, dimension( *
) WORK, double precision, dimension( * ) RWORK)
ZLA_GERCOND_X computes the infinity norm condition number of op(A)*diag(x) for general
matrices.
Purpose:
ZLA_GERCOND_X computes the infinity norm condition number of
op(A) * diag(X) where X is a COMPLEX*16 vector.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate Transpose = Transpose)
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by ZGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by ZGETRF; row i of the matrix was interchanged
with row IPIV(i).
X
X is COMPLEX*16 array, dimension (N)
The vector X in the formula op(A) * diag(X).
INFO
INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.
WORK
WORK is COMPLEX*16 array, dimension (2*N).
Workspace.
RWORK
RWORK is DOUBLE PRECISION array, dimension (N).
Workspace.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zla_gerfsx_extended (integer PREC_TYPE, integer TRANS_TYPE, integer N, integer
NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF,
integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, double precision, dimension( *
) C, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldy, * ) Y,
integer LDY, double precision, dimension( * ) BERR_OUT, integer N_NORMS, double precision,
dimension( nrhs, * ) ERRS_N, double precision, dimension( nrhs, * ) ERRS_C, complex*16,
dimension( * ) RES, double precision, dimension( * ) AYB, complex*16, dimension( * ) DY,
complex*16, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double
precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO)
ZLA_GERFSX_EXTENDED
Purpose:
ZLA_GERFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by ZGERFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERRS_N
and ERRS_C for details of the error bounds. Note that this
subroutine is only responsible for setting the second fields of
ERRS_N and ERRS_C.
Parameters
PREC_TYPE
PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X' or 'E': Extra
TRANS_TYPE
TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and T
= 'N': No transpose
= 'T': Transpose
= 'C': Conjugate transpose
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns of the
matrix B.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by ZGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV
IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by ZGETRF; row i of the matrix was interchanged
with row IPIV(i).
COLEQU
COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.
C
C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. If C is input, each element of C should be a power
of the radix to ensure a reliable solution and error estimates.
Scaling by powers of the radix does not cause rounding errors unless
the result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.
B
B is COMPLEX*16 array, dimension (LDB,NRHS)
The right-hand-side matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Y
Y is COMPLEX*16 array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by ZGETRS.
On exit, the improved solution matrix Y.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT
BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by ZLA_LIN_BERR.
N_NORMS
N_NORMS is INTEGER
Determines which error bounds to return (see ERRS_N
and ERRS_C).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.
ERRS_N
ERRS_N is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERRS_N(i,:) corresponds to the ith
right-hand side.
The second index in ERRS_N(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
ERRS_C
ERRS_C is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERRS_C is not accessed. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERRS_C(i,:) corresponds to the ith
right-hand side.
The second index in ERRS_C(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.
RES
RES is COMPLEX*16 array, dimension (N)
Workspace to hold the intermediate residual.
AYB
AYB is DOUBLE PRECISION array, dimension (N)
Workspace.
DY
DY is COMPLEX*16 array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is COMPLEX*16 array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.
RCOND
RCOND is DOUBLE PRECISION
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.
ITHRESH
ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For 'aggressive' set to 100 to
permit convergence using approximate factorizations or
factorizations other than LU. If the factorization uses a
technique other than Gaussian elimination, the guarantees in
ERRS_N and ERRS_C may no longer be trustworthy.
RTHRESH
RTHRESH is DOUBLE PRECISION
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.
DZ_UB
DZ_UB is DOUBLE PRECISION
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.
IGNORE_CWISE
IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..
INFO
INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to ZGETRS had an illegal
value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zla_gerpvgrw (integer N, integer NCOLS, complex*16, dimension( lda,
* ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF)
ZLA_GERPVGRW multiplies a square real matrix by a complex matrix.
Purpose:
ZLA_GERPVGRW computes the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute element" norm is used. If this is
much less than 1, the stability of the LU factorization of the
(equilibrated) matrix A could be poor. This also means that the
solution X, estimated condition numbers, and error bounds could be
unreliable.
Parameters
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NCOLS
NCOLS is INTEGER
The number of columns of the matrix A. NCOLS >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by ZGETRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
recursive subroutine zlaqz0 (character, intent(in) WANTS, character, intent(in) WANTQ,
character, intent(in) WANTZ, integer, intent(in) N, integer, intent(in) ILO, integer,
intent(in) IHI, complex*16, dimension( lda, * ), intent(inout) A, integer, intent(in) LDA,
complex*16, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB, complex*16,
dimension( * ), intent(inout) ALPHA, complex*16, dimension( * ), intent(inout) BETA,
complex*16, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ,
complex*16, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, complex*16,
dimension( * ), intent(inout) WORK, integer, intent(in) LWORK, double precision,
dimension( * ), intent(out) RWORK, integer, intent(in) REC, integer, intent(out) INFO)
ZLAQZ0
Purpose:
ZLAQZ0 computes the eigenvalues of a real matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a real matrix pair (A,B):
A = Q1*H*Z1**H, B = Q1*T*Z1**H,
as computed by ZGGHRD.
If JOB='S', then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,
H = Q*S*Z**H, T = Q*P*Z**H,
where Q and Z are unitary matrices, P and S are an upper triangular
matrices.
Optionally, the unitary matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
unitary matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the unitary factors from the
generalized Schur factorization of (A,B):
A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
complex and beta real.
If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
generalized nonsymmetric eigenvalue problem (GNEP)
A*x = lambda*B*x
and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
alternate form of the GNEP
mu*A*y = B*y.
Eigenvalues can be read directly from the generalized Schur
form:
alpha = S(i,i), beta = P(i,i).
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
Ref: B. Kagstrom, D. Kressner, "Multishift Variants of the QZ
Algorithm with Aggressive Early Deflation", SIAM J. Numer.
Anal., 29(2006), pp. 199--227.
Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril "A multishift,
multipole rational QZ method with agressive early deflation"
Parameters
WANTS
WANTS is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Compute eigenvalues and the Schur form.
WANTQ
WANTQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (A,B) is returned;
= 'V': Q must contain an unitary matrix Q1 on entry and
the product Q1*Q is returned.
WANTZ
WANTZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (A,B) is returned;
= 'V': Z must contain an unitary matrix Z1 on entry and
the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of A which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
A
A is COMPLEX*16 array, dimension (LDA, N)
On entry, the N-by-N upper Hessenberg matrix A.
On exit, if JOB = 'S', A contains the upper triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal blocks of A match those of S, but
the rest of A is unspecified.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B
B is COMPLEX*16 array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, if JOB = 'S', B contains the upper triangular
matrix P from the generalized Schur factorization;
If JOB = 'E', the diagonal blocks of B match those of P, but
the rest of B is unspecified.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHA
ALPHA is COMPLEX*16 array, dimension (N)
Each scalar alpha defining an eigenvalue
of GNEP.
BETA
BETA is COMPLEX*16 array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = ALPHA(j) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha. Since either lambda or mu may overflow,
they should not, in general, be computed.
Q
Q is COMPLEX*16 array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the unitary matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the unitary matrix of left Schur
vectors of (A,B), and if COMPQ = 'V', the unitary matrix
of left Schur vectors of (A,B).
Not referenced if COMPQ = 'N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z
Z is COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the unitary matrix of
right Schur vectors of (H,T), and if COMPZ = 'V', the
unitary matrix of right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
RWORK
RWORK is DOUBLE PRECISION array, dimension (N)
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
REC
REC is INTEGER
REC indicates the current recursion level. Should be set
to 0 on first call.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (A,B) is not
in Schur form, but ALPHA(i) and
BETA(i), i=INFO+1,...,N should be correct.
Author
Thijs Steel, KU Leuven
Date
May 2020
subroutine zlaqz1 (logical, intent(in) ILQ, logical, intent(in) ILZ, integer, intent(in) K,
integer, intent(in) ISTARTM, integer, intent(in) ISTOPM, integer, intent(in) IHI,
complex*16, dimension( lda, * ) A, integer, intent(in) LDA, complex*16, dimension( ldb, *
) B, integer, intent(in) LDB, integer, intent(in) NQ, integer, intent(in) QSTART,
complex*16, dimension( ldq, * ) Q, integer, intent(in) LDQ, integer, intent(in) NZ,
integer, intent(in) ZSTART, complex*16, dimension( ldz, * ) Z, integer, intent(in) LDZ)
ZLAQZ1
Purpose:
ZLAQZ1 chases a 1x1 shift bulge in a matrix pencil down a single position
Parameters
ILQ
ILQ is LOGICAL
Determines whether or not to update the matrix Q
ILZ
ILZ is LOGICAL
Determines whether or not to update the matrix Z
K
K is INTEGER
Index indicating the position of the bulge.
On entry, the bulge is located in
(A(k+1,k),B(k+1,k)).
On exit, the bulge is located in
(A(k+2,k+1),B(k+2,k+1)).
ISTARTM
ISTARTM is INTEGER
ISTOPM
ISTOPM is INTEGER
Updates to (A,B) are restricted to
(istartm:k+2,k:istopm). It is assumed
without checking that istartm <= k+1 and
k+2 <= istopm
IHI
IHI is INTEGER
A
A is COMPLEX*16 array, dimension (LDA,N)
LDA
LDA is INTEGER
The leading dimension of A as declared in
the calling procedure.
B
B is COMPLEX*16 array, dimension (LDB,N)
LDB
LDB is INTEGER
The leading dimension of B as declared in
the calling procedure.
NQ
NQ is INTEGER
The order of the matrix Q
QSTART
QSTART is INTEGER
Start index of the matrix Q. Rotations are applied
To columns k+2-qStart:k+3-qStart of Q.
Q
Q is COMPLEX*16 array, dimension (LDQ,NQ)
LDQ
LDQ is INTEGER
The leading dimension of Q as declared in
the calling procedure.
NZ
NZ is INTEGER
The order of the matrix Z
ZSTART
ZSTART is INTEGER
Start index of the matrix Z. Rotations are applied
To columns k+1-qStart:k+2-qStart of Z.
Z
Z is COMPLEX*16 array, dimension (LDZ,NZ)
LDZ
LDZ is INTEGER
The leading dimension of Q as declared in
the calling procedure.
Author
Thijs Steel, KU Leuven
Date
May 2020
recursive subroutine zlaqz2 (logical, intent(in) ILSCHUR, logical, intent(in) ILQ, logical,
intent(in) ILZ, integer, intent(in) N, integer, intent(in) ILO, integer, intent(in) IHI,
integer, intent(in) NW, complex*16, dimension( lda, * ), intent(inout) A, integer,
intent(in) LDA, complex*16, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB,
complex*16, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ,
complex*16, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, integer,
intent(out) NS, integer, intent(out) ND, complex*16, dimension( * ), intent(inout) ALPHA,
complex*16, dimension( * ), intent(inout) BETA, complex*16, dimension( ldqc, * ) QC,
integer, intent(in) LDQC, complex*16, dimension( ldzc, * ) ZC, integer, intent(in) LDZC,
complex*16, dimension( * ) WORK, integer, intent(in) LWORK, double precision, dimension( *
) RWORK, integer, intent(in) REC, integer, intent(out) INFO)
ZLAQZ2
Purpose:
ZLAQZ2 performs AED
Parameters
ILSCHUR
ILSCHUR is LOGICAL
Determines whether or not to update the full Schur form
ILQ
ILQ is LOGICAL
Determines whether or not to update the matrix Q
ILZ
ILZ is LOGICAL
Determines whether or not to update the matrix Z
N
N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of (A,B) which
are to be normalized
NW
NW is INTEGER
The desired size of the deflation window.
A
A is COMPLEX*16 array, dimension (LDA, N)
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B
B is COMPLEX*16 array, dimension (LDB, N)
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
Q
Q is COMPLEX*16 array, dimension (LDQ, N)
LDQ
LDQ is INTEGER
Z
Z is COMPLEX*16 array, dimension (LDZ, N)
LDZ
LDZ is INTEGER
NS
NS is INTEGER
The number of unconverged eigenvalues available to
use as shifts.
ND
ND is INTEGER
The number of converged eigenvalues found.
ALPHA
ALPHA is COMPLEX*16 array, dimension (N)
Each scalar alpha defining an eigenvalue
of GNEP.
BETA
BETA is COMPLEX*16 array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = ALPHA(j) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha. Since either lambda or mu may overflow,
they should not, in general, be computed.
QC
QC is COMPLEX*16 array, dimension (LDQC, NW)
LDQC
LDQC is INTEGER
ZC
ZC is COMPLEX*16 array, dimension (LDZC, NW)
LDZC
LDZ is INTEGER
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
RWORK
RWORK is DOUBLE PRECISION array, dimension (N)
REC
REC is INTEGER
REC indicates the current recursion level. Should be set
to 0 on first call.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Thijs Steel, KU Leuven
Date
May 2020
subroutine zlaqz3 (logical, intent(in) ILSCHUR, logical, intent(in) ILQ, logical, intent(in)
ILZ, integer, intent(in) N, integer, intent(in) ILO, integer, intent(in) IHI, integer,
intent(in) NSHIFTS, integer, intent(in) NBLOCK_DESIRED, complex*16, dimension( * ),
intent(inout) ALPHA, complex*16, dimension( * ), intent(inout) BETA, complex*16,
dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, complex*16, dimension( ldb,
* ), intent(inout) B, integer, intent(in) LDB, complex*16, dimension( ldq, * ),
intent(inout) Q, integer, intent(in) LDQ, complex*16, dimension( ldz, * ), intent(inout)
Z, integer, intent(in) LDZ, complex*16, dimension( ldqc, * ), intent(inout) QC, integer,
intent(in) LDQC, complex*16, dimension( ldzc, * ), intent(inout) ZC, integer, intent(in)
LDZC, complex*16, dimension( * ), intent(inout) WORK, integer, intent(in) LWORK, integer,
intent(out) INFO)
ZLAQZ3
Purpose:
ZLAQZ3 Executes a single multishift QZ sweep
Parameters
ILSCHUR
ILSCHUR is LOGICAL
Determines whether or not to update the full Schur form
ILQ
ILQ is LOGICAL
Determines whether or not to update the matrix Q
ILZ
ILZ is LOGICAL
Determines whether or not to update the matrix Z
N
N is INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
NSHIFTS
NSHIFTS is INTEGER
The desired number of shifts to use
NBLOCK_DESIRED
NBLOCK_DESIRED is INTEGER
The desired size of the computational windows
ALPHA
ALPHA is COMPLEX*16 array. SR contains
the alpha parts of the shifts to use.
BETA
BETA is COMPLEX*16 array. SS contains
the scale of the shifts to use.
A
A is COMPLEX*16 array, dimension (LDA, N)
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B
B is COMPLEX*16 array, dimension (LDB, N)
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
Q
Q is COMPLEX*16 array, dimension (LDQ, N)
LDQ
LDQ is INTEGER
Z
Z is COMPLEX*16 array, dimension (LDZ, N)
LDZ
LDZ is INTEGER
QC
QC is COMPLEX*16 array, dimension (LDQC, NBLOCK_DESIRED)
LDQC
LDQC is INTEGER
ZC
ZC is COMPLEX*16 array, dimension (LDZC, NBLOCK_DESIRED)
LDZC
LDZ is INTEGER
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Thijs Steel, KU Leuven
Date
May 2020
subroutine zlaunhr_col_getrfnp (integer M, integer N, complex*16, dimension( lda, * ) A,
integer LDA, complex*16, dimension( * ) D, integer INFO)
ZLAUNHR_COL_GETRFNP
Purpose:
ZLAUNHR_COL_GETRFNP computes the modified LU factorization without
pivoting of a complex general M-by-N matrix A. The factorization has
the form:
A - S = L * U,
where:
S is a m-by-n diagonal sign matrix with the diagonal D, so that
D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
i-1 steps of Gaussian elimination. This means that the diagonal
element at each step of "modified" Gaussian elimination is
at least one in absolute value (so that division-by-zero not
not possible during the division by the diagonal element);
L is a M-by-N lower triangular matrix with unit diagonal elements
(lower trapezoidal if M > N);
and U is a M-by-N upper triangular matrix
(upper trapezoidal if M < N).
This routine is an auxiliary routine used in the Householder
reconstruction routine ZUNHR_COL. In ZUNHR_COL, this routine is
applied to an M-by-N matrix A with orthonormal columns, where each
element is bounded by one in absolute value. With the choice of
the matrix S above, one can show that the diagonal element at each
step of Gaussian elimination is the largest (in absolute value) in
the column on or below the diagonal, so that no pivoting is required
for numerical stability [1].
For more details on the Householder reconstruction algorithm,
including the modified LU factorization, see [1].
This is the blocked right-looking version of the algorithm,
calling Level 3 BLAS to update the submatrix. To factorize a block,
this routine calls the recursive routine ZLAUNHR_COL_GETRFNP2.
[1] "Reconstructing Householder vectors from tall-skinny QR",
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A-S=L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D
D is COMPLEX*16 array, dimension min(M,N)
The diagonal elements of the diagonal M-by-N sign matrix S,
D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be
only ( +1.0, 0.0 ) or (-1.0, 0.0 ).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
recursive subroutine zlaunhr_col_getrfnp2 (integer M, integer N, complex*16, dimension( lda, *
) A, integer LDA, complex*16, dimension( * ) D, integer INFO)
ZLAUNHR_COL_GETRFNP2
Purpose:
ZLAUNHR_COL_GETRFNP2 computes the modified LU factorization without
pivoting of a complex general M-by-N matrix A. The factorization has
the form:
A - S = L * U,
where:
S is a m-by-n diagonal sign matrix with the diagonal D, so that
D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
i-1 steps of Gaussian elimination. This means that the diagonal
element at each step of "modified" Gaussian elimination is at
least one in absolute value (so that division-by-zero not
possible during the division by the diagonal element);
L is a M-by-N lower triangular matrix with unit diagonal elements
(lower trapezoidal if M > N);
and U is a M-by-N upper triangular matrix
(upper trapezoidal if M < N).
This routine is an auxiliary routine used in the Householder
reconstruction routine ZUNHR_COL. In ZUNHR_COL, this routine is
applied to an M-by-N matrix A with orthonormal columns, where each
element is bounded by one in absolute value. With the choice of
the matrix S above, one can show that the diagonal element at each
step of Gaussian elimination is the largest (in absolute value) in
the column on or below the diagonal, so that no pivoting is required
for numerical stability [1].
For more details on the Householder reconstruction algorithm,
including the modified LU factorization, see [1].
This is the recursive version of the LU factorization algorithm.
Denote A - S by B. The algorithm divides the matrix B into four
submatrices:
[ B11 | B12 ] where B11 is n1 by n1,
B = [ -----|----- ] B21 is (m-n1) by n1,
[ B21 | B22 ] B12 is n1 by n2,
B22 is (m-n1) by n2,
with n1 = min(m,n)/2, n2 = n-n1.
The subroutine calls itself to factor B11, solves for B21,
solves for B12, updates B22, then calls itself to factor B22.
For more details on the recursive LU algorithm, see [2].
ZLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked
routine ZLAUNHR_COL_GETRFNP, which uses blocked code calling
Level 3 BLAS to update the submatrix. However, ZLAUNHR_COL_GETRFNP2
is self-sufficient and can be used without ZLAUNHR_COL_GETRFNP.
[1] "Reconstructing Householder vectors from tall-skinny QR",
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.
[2] "Recursion leads to automatic variable blocking for dense linear
algebra algorithms", F. Gustavson, IBM J. of Res. and Dev.,
vol. 41, no. 6, pp. 737-755, 1997.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A-S=L*U; the unit diagonal elements of L are not stored.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D
D is COMPLEX*16 array, dimension min(M,N)
The diagonal elements of the diagonal M-by-N sign matrix S,
D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be
only ( +1.0, 0.0 ) or (-1.0, 0.0 ).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
subroutine ztgevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer
N, complex*16, dimension( lds, * ) S, integer LDS, complex*16, dimension( ldp, * ) P,
integer LDP, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension(
ldvr, * ) VR, integer LDVR, integer MM, integer M, complex*16, dimension( * ) WORK, double
precision, dimension( * ) RWORK, integer INFO)
ZTGEVC
Purpose:
ZTGEVC computes some or all of the right and/or left eigenvectors of
a pair of complex matrices (S,P), where S and P are upper triangular.
Matrix pairs of this type are produced by the generalized Schur
factorization of a complex matrix pair (A,B):
A = Q*S*Z**H, B = Q*P*Z**H
as computed by ZGGHRD + ZHGEQZ.
The right eigenvector x and the left eigenvector y of (S,P)
corresponding to an eigenvalue w are defined by:
S*x = w*P*x, (y**H)*S = w*(y**H)*P,
where y**H denotes the conjugate tranpose of y.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal elements of S and P.
This routine returns the matrices X and/or Y of right and left
eigenvectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices.
If Q and Z are the unitary factors from the generalized Schur
factorization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B).
Parameters
SIDE
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT.
SELECT
SELECT is LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors to be
computed. The eigenvector corresponding to the j-th
eigenvalue is computed if SELECT(j) = .TRUE..
Not referenced if HOWMNY = 'A' or 'B'.
N
N is INTEGER
The order of the matrices S and P. N >= 0.
S
S is COMPLEX*16 array, dimension (LDS,N)
The upper triangular matrix S from a generalized Schur
factorization, as computed by ZHGEQZ.
LDS
LDS is INTEGER
The leading dimension of array S. LDS >= max(1,N).
P
P is COMPLEX*16 array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur
factorization, as computed by ZHGEQZ. P must have real
diagonal elements.
LDP
LDP is INTEGER
The leading dimension of array P. LDP >= max(1,N).
VL
VL is COMPLEX*16 array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the unitary matrix Q
of left Schur vectors returned by ZHGEQZ).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VL, in the same order as their eigenvalues.
Not referenced if SIDE = 'R'.
LDVL
LDVL is INTEGER
The leading dimension of array VL. LDVL >= 1, and if
SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N.
VR
VR is COMPLEX*16 array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Q (usually the unitary matrix Z
of right Schur vectors returned by ZHGEQZ).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Z*X;
if HOWMNY = 'S', the right eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VR, in the same order as their eigenvalues.
Not referenced if SIDE = 'L'.
LDVR
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N.
MM
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
is set to N. Each selected eigenvector occupies one column.
WORK
WORK is COMPLEX*16 array, dimension (2*N)
RWORK
RWORK is DOUBLE PRECISION array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine ztgexc (logical WANTQ, logical WANTZ, integer N, complex*16, dimension( lda, * ) A,
integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldq, *
) Q, integer LDQ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer IFST, integer
ILST, integer INFO)
ZTGEXC
Purpose:
ZTGEXC reorders the generalized Schur decomposition of a complex
matrix pair (A,B), using an unitary equivalence transformation
(A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with
row index IFST is moved to row ILST.
(A, B) must be in generalized Schur canonical form, that is, A and
B are both upper triangular.
Optionally, the matrices Q and Z of generalized Schur vectors are
updated.
Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H
Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H
Parameters
WANTQ
WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
WANTZ
WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the upper triangular matrix A in the pair (A, B).
On exit, the updated matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is COMPLEX*16 array, dimension (LDB,N)
On entry, the upper triangular matrix B in the pair (A, B).
On exit, the updated matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q
Q is COMPLEX*16 array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., the unitary matrix Q.
On exit, the updated matrix Q.
If WANTQ = .FALSE., Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1;
If WANTQ = .TRUE., LDQ >= N.
Z
Z is COMPLEX*16 array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., the unitary matrix Z.
On exit, the updated matrix Z.
If WANTZ = .FALSE., Z is not referenced.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1;
If WANTZ = .TRUE., LDZ >= N.
IFST
IFST is INTEGER
ILST
ILST is INTEGER
Specify the reordering of the diagonal blocks of (A, B).
The block with row index IFST is moved to row ILST, by a
sequence of swapping between adjacent blocks.
INFO
INFO is INTEGER
=0: Successful exit.
<0: if INFO = -i, the i-th argument had an illegal value.
=1: The transformed matrix pair (A, B) would be too far
from generalized Schur form; the problem is ill-
conditioned. (A, B) may have been partially reordered,
and ILST points to the first row of the current
position of the block being moved.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
87 Umea, Sweden.
References:
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real
Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra
for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a
Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software,
Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the
Generalized Sylvester Equation and Estimating the Separation between Regular Matrix
Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901
87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To
appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
Author
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